http://www.ck12.org Chapter 2. Polynomials and Rational Functions
Notice on the right portion of the graph the curve seems to stay on thex-axis. In fact it does go slightly above and
belowx-axis, crossing through it at (1, 0), (2, 0) and (3, 0) before starting to increase.
Example C
Estimate a function that would have the following graphical characteristics:
Solution: First think about the vertical asymptotes and how they affect the equation of the function. Then consider
zeroes and holes, and the way the graph looks at these places. Finally, use they-intercept to refine your equation.
- The function has two vertical asymptotes atx=− 2 ,1 so the denominator must have the factors(x+ 2 )(x− 1 ).
- There is one zero atx=−1, so the numerator must have a factor of(x+ 1 ).
- There are two holes that appear to override zeroes which means the numerator and denominator must have the
factors(x+ 3 )and(x− 4 ). - Because the graph goes from above thex-axis to below thex-axis atx=−3, the degree of the exponent of
the(x+ 3 )factor must be ultimately odd. - Because the graph stays above thex-axis before and afterx=4, the degree of the(x− 4 )factor must be
ultimately even.
A good estimate for the function is:
f(x) = (x+^1 )(x+^3 )(^4) (x− 4 ) 3
(x+ 2 )(x− 1 )(x+ 3 )(x− 4 )
This function has all the basic characteristics, however it isn’t scaled properly. Whenx=0 this function has a
y-intercept of -216 when it should be about -2. Thus you must divide by 108 so that theyintercept matches. Here
is a better estimate for the function:
f(x) = (x+^1 )(x+^3 )
(^4) (x− 4 ) 3
108 (x+ 2 )(x− 1 )(x+ 3 )(x− 4 )
Concept Problem Revisited
Computers can graph rational functions more accurately than people. However, computers may not be able to
explain why a function behaves in certain ways. By being a detective and looking for clues in the equation of a
function, you are applying high level analytical skills and powers of deduction. These analytical skills are vastly
more important and transferrable than the specific techniques involved with rational functions.